冯·诺依曼代数作为约化扭曲群胚 $C^*$-代数
Von Neumann algebras as reduced twisted groupoid $C^*$-algebras
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中文总结 AI 辅助
本文刻画了同构于约化扭曲群胚$C^*$-代数的冯·诺依曼代数，证明其为次齐次代数，并引入受控传播障碍，排除$B(H)$及Calkin代数的嵌入。
中文摘要 AI 辅助
我们刻画了作为 $C^*$-代数同构于具有全支撑连续 Haar 系的局部紧 Hausdorff 群胚的约化扭曲 $C^*$-代数的冯·诺依曼代数。它们恰好是次齐次冯·诺依曼代数，等价于交换冯·诺依曼代数上的矩阵代数的有限乘积。群胚总可以被选为紧、主且étale的，具有平凡扭曲和计数 Haar 系。我们还证明，对于此类群胚，全约化或约化扭曲代数是单位的当且仅当群胚是étale且单位空间紧致。这一准则将一般 Haar 系的分类归结为étale情形。进一步的障碍来自受控传播，它通过忠实表示定义到一致稀疏矩阵的范数闭包中。每个约化扭曲étale群胚代数及其 Borel 完备化都具有受控传播，包括对具有局部紧 Hausdorff 单位空间的非 Hausdorff 群胚。对于每个无限维 Hilbert 空间 $H$，从 $B(H)$ 的非零商到具有受控传播的代数的每个 $*$-同态都是零。特别地，$B(H)$ 和 Calkin 代数都不能嵌入到这些约化或 Borel 群胚代数中的任何一个。我们还证明，每个具有受控传播的冯·诺依曼代数都是有限的。不需要可分性或可数性假设。
英文摘要
We characterize the von Neumann algebras that are isomorphic, as $C^*$-algebras, to reduced twisted $C^*$-algebras of locally compact Hausdorff groupoids equipped with continuous Haar systems of full support. They are exactly the subhomogeneous von Neumann algebras, equivalently finite products of matrix algebras over abelian von Neumann algebras. The groupoid can always be chosen compact, principal, and étale, with trivial twist and counting Haar system. We also prove that, for a groupoid in this class, the full or reduced twisted algebra is unital if and only if the groupoid is étale with compact unit space. This criterion reduces the classification for general Haar systems to the étale case. A further obstruction comes from controlled propagation, defined through faithful representations into the norm closure of uniformly sparse matrices. Every reduced twisted étale groupoid algebra and its Borel completion have controlled propagation, including for non-Hausdorff groupoids with locally compact Hausdorff unit space. For every infinite-dimensional Hilbert space $H$, every $*$-homomorphism from a nonzero quotient of $B(H)$ into an algebra with controlled propagation is zero. In particular, neither $B(H)$ nor the Calkin algebra embeds into any of these reduced or Borel groupoid algebras. We also prove that every von Neumann algebra with controlled propagation is finite. No separability or countability assumptions are required.
发表机构
- Universidade Federal de Santa Catarina（圣卡塔琳娜联邦大学）
- Instituto Superior Técnico, University of Lisbon（里斯本大学高等技术学院）
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